橡胶衬套材料的超弹性力学行为研究

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Sep．2013机 床 与 液 压Hydromechatronics Engineering V01．41 No．18DOI：10．3969／j．issn．1001—3881．2013．18．004A Study on M echanical Behavior of Hyperelastic M aterialfor Rubber BushingLEI Gang ，CHEN Qian ， CHEN Bao 一．ZHAO Peng． a．Key Laboratory of Manufactory and Test Techniques of Automobile Parts，Ministry Education；b．School of Mechanical Enginering，Chongqing Unive~ity of Technology，Chongqing 400054，China；2．School ofMechanical Enginering，Southwest flaotong Unive~ity，Chengdu 610031，China1．IntroductionsAbstract：The rubber bushings of vehicle suspension，made of mainly rubber materials and ametal or plastic sleeve vulcanized to rubber，are indispensable components impacting vehicle ridecomfort，handle stability and NVH etc．In this paper，the characteristics of bushing rubbers wereresearched to intensive analyze how the rubber bushing afects vehicle performances．A methodis introduced to identify and fit the hyperelastic coeficients of the selected constitutive modeI by u-sing three fypes of material test data in ABAQUS and HyperStudy，and a more high-accurate Fi-nite Element(FE)modeI of rubber bushing was built to simulate the radial and axiaI stifnesscuwes by seRing actual constraint conditions．A comparison was carried out between the simula-tion results and experimentaI results．The conclusion shows that the Van der Waals modeIis Iike—ly a clearer superiority representation than other constitutive models，which can be used for thesimulation to beRer fit the hyperelastic coeficients.

Key words：rubber bushing，constitutive model，material test，parameter identificationSuspension is an important sub—system of anyvehicle，and rubber bushings are important compo—nents of a suspension[1]．As a typical rubber—likematerial with hyperelastic properties，rubber bushinghas some special functions， such as transferringforces，enabling defined movements，noise isolation，and vibration damping[2]．The mechanical behaviorof rubber—like materials is usualy characterized by astrain energy density function．The coeficients deft—Received：2013—03—28Sponsored by National Natural Science Foundation of China(5 1205433)，and by National Natural Science Foundation ofChongqing(cstc201 ljA60003)，and by 2012 Open Founda—tion of Key Laboratory of Manufactory and Test Techniques ofAutomobile Pans，Ministry Education，Chongqing Universityof Technology.

LEI Gang．Professor．E—mail：leigang### cqut．edu．cnning the density function are considered as materialparameters that need to be identifed．However，thecomplex molecular structure and nonlinear propertiesof material and geometry make it complicated to de—scribe its behavior． The published literatures haveput forward some ways to identify material parametersof different constitutive equations from different as-pects[3—4]．But each constitutive relationship hasitself respective characteristics and applicable scope.

So selection of constitutive equation for rubber—likematerial plays significant role in Finite Element AnaJ—ysis(FEA).

In this paper，the identification of the parame—ters defining the material properties required was test—ed in homogeneous deformation modes by using tradi—tional laboratory technique，such as uniaxial tension，biaxial tension，and planar shear．Depending on theavailable test data，the coeficients of diferent consti—tutive equations were calculated in ABAQUS and Hy.

perStudy，and then the fitting coeficients were COB—20 Hydromechatronics Engineerin
—
g
— pared with the test data，and further the coefficientswere employed in establishing the FE model of therubber bushing．the radial and axial stiffness curvescan be gotten by simulation analysis．By comparingthe simulation results with the test results， betterconstitutive equation for the rubber bushing can bedecided.

2．The material tests of rubber bushingThe mechanical behavior of a bushing rubber isdeftned as a stress．strain relationship which derivedfrom the material test．The material coeficients in thestrain energy functions can be determined from thecurve fiting of stress．strain test data．There are sev·eral different types of tests，including simple tension，equi—biaxial tension and pure shear tests．In general，a combination of simple tension，equi—biaxial tensionand pure shear tests are used to determine the materi—al coeficient．The classical Mooney—Rivlin and Og—den model are an example of a hyperelastic modelthat is implemented in FEA.

The rubber material property．which is essentialin FEA．is expressed with the coeficient values ofstrain energy function and these values are deter—mined by fitting stress — strain data obtained fromthe material tests under various load conditions intothe stress．strain curve induced from strain energyfuBetion．And it is determined to minimize the differ—erices between the test values and calculated values.

Therefore，the material properties are analyzed andthe nonlinear material coeficients are determined，which is necessary in finite element analysis。by con—ducting uniaxial，equi-biaxial tension and pure sheartests『5].

3．Constitutive equations of rubber3．1．Hyperelastic material modelsBesides being hyperelastic and incompressible，rubber is usually treated as an isotropic materia1．Hy—perelastic materials are described in terms of a‘strain energy potential’，which defines the strainenergy stored in the material per unit of refe rence vol—ume as a function of the deformation at that point inthe materia1． Each constitutive model is a specialform of the strain energy potential[6]．The strain en—ergy form can be classified into the phenomenol0gicaldescription and the statistical mechanics．based de·-scription．In present，the maturity constitutive mod—els are summarized in Tab．1.

Tab．1 Popular constitutive models for rubber-likematerials in FEM analysisConstitutive models Number of matedalparametersFor isotropic material，the Polynomial strain en—ergy potential is expressed as below[7].

，v= ∑c (7 一3) (I2—3) +i+J=1N 1∑去( 一1) (1)where U denotes the strain energy per unit referencevolume，N are known as the order of the polynomial，D describe the compressibility，and the material isincompressible if all values of D are equal to zero，Iland，1 denotes the first and second invariants of devi-atoric part of fight-Cauchy-Green deform ation tensorrespectivelv，了is the ratio of deformed volume overthe undeformed volume of materials.

By setting N = 1．the Mooney—Rivlin model isdefined．If seting G = 0 for J≠ 0，the reducedpolynomial model is obtained．The simplest hypere-lastic model and the Neo—Hookean model can be ob—tained as the reduced polynomial with N =1．In addi—tion．if N is set to 3，the Yeoh model is obtained.

The Ogden strain energy potential is expressedin terms of the principal stretches(A1，A2，A 3).

Thus，the constitutive equation of Ogden takes thefolowing form[8].

U=∑ 2t-／-,~—t (A
1od+A； +A； 一3)+妻
i=1c 7 (2)The Ogden form Eq．(2)assumes that rubber iselastic and isotropic so that U may be expressed as afunction of the strain invariants．Note that as requiredbv the definition of strain energy potential，Ogden’sstrain energy vanishes when the material is in its nat—ural，undeformed state人1=人2=A 3 1[9].

LEI Gang，et al：A Study on Mechanical Behavior of Hyperelastic Materialfor Rubber BushingThe Arruda—Boyce form of strain energy potentialis written as below.

= 善 Cii 1c +吉( t，) =^m 、
(3)The Arruda·—Boyce model is also called the eight-chain mode1．The values for C1? C5 is defined bystatistical thermodynamics．Where denotes the ini—tial shear modulus of the materia1．The parameter Arepresents the locking stretch．The volume modulusis k0=2／D．The hyperelastic constitutive models canbe obtained with little practical data on the account ofonly two unknown parameter.

The Van der Waals strain energy model has thefolowing form [10].

u= { -3)【ln(1_ 了2 a( ) )+古(Where／=(1一卢)I1+ ，一 1n．，1for the special polynomial models (Yeoh，Neo—Hookean and Mooney．Rivlin models)were comparedwith the test data shown in Fig．1.

For the polynomial models． the case N = 1(Mooney—Rivlin)gives a reasonable fiting at lowstrains，but is unable to reproduce the stiffening re—sponse of the rubber at higher strains．Similar obser-vations were applied to the reduced polynomial withN=1(neo-Hookean)and N=3(Yeoh)．The neo—Hookean model ofers only a linear dependence of thefirst invariant，which fails to provide an accurate rep—resentation of the upturn．In contrast。the three．orderreduced polynomial(Yeoh)provides a more accuraterepresentation．which has five coeficients．The coef-ficients of these strain energy models were listed inTah．2.

Tab．2 The coeficients of polynomial models(4) Constitutive Equationsis a linear parameterwhich couplell and I2to I．and T／3．2．Hyperelastic coefficients fitting procedureA11 three types of test data are used simuhane.

ously in fitting the hyperelastic coeficients of rubber—like materials．The polynomial potential is linear interm s of the constants C ，；therefore，a linear least.

squares procedure can be used．To the Ogden，theArruda—Boyce and the Van der Waals potentia1．someof their coefficients are nonlinear，thus necessitatingthe use of a nonlinear least-squares procedure[11].

Those parameters are effectively evaluated in nonlin.

ear finite element of Abaqus.

3．2．1．Fitting for the polynomial modelsThe Abaqus nominal stress—nominal strain resultsNominai Srain(a)Uniaxial tension3 O日 2 52 0器至1．5∞i 1．0oz 0
．
50Polynomial N ：1(Mooney—Rivlin)Cl0=0．562 1，C0】=4．751e一02Reduc
，
e Po 啪 i G。：0．604 1
N 1 N = f eo Hooke)Reduce PolynomialN=3(Yeoh)C10=0．638 7，C20= 一7．288e一02，C∞=2．191e一023．2．2．Fitting for the Ogden．Arruda—Boyce and Vander Waals modelsAll types of test data are used in fitting the hy—perelastic coefficients for order N =1．N =3 Ogdenmodels．Arruda．Boyce and Van der Waals models.

The Abaqus simulation results were compared withthe test data shown in Fig．2.

0 0 2 0 4 0 6 0 8 1 0 1 2Nominal Srain(b)Biaxial tension3 O2 52 0器至1 5兽； 1 0主 0 50Fig．1 Evaluate results for polynomial models0 2 0 4 0 6 0 8 1 0 1 2NominaI Srain(c)Planar shear，s∞ ∞ 口 g oZ22 Hydromechatronics Engineering3 O2 52 Ol 51 00 50O 2 O 4 O 6 O．8 l 0 l 2 0 0 2 O 4 O 6 0 8 1 0 1 2Nominal Srain Nominal Srain(a)Uniaxial tension (b)Biaxial tension3 02 52 01 5l O0 5OFig．2 Evaluate results of Ogden，Arruda—Boyce and Van der Waals modelsIn Fig．2．the case N =1 of Ogden model canreproduce the stiffening response of the rubber at lowstrains but distorts at higher strains．Reverse observa—tions applied to the three—order Ogden strain energymodel provides a more accurate representation．whichhas six coeficients．If the test data are more pro—nounced in the S．shape．as is common for filled rub.

bers，the Van der Waals model is likely to show aneven clearer superiority．Since the additional parame—ters creating enhanced flexibility does not exhibit anyinstability in representing complex stress—strainCHIVES．While the Arruda．Boyce model gives an un.

desirable outcome which means that this model canonly fit tightly at very smallThe fited coemcientsand 4range of strain.

are shown in Tab． 3Tab．3 The coeficients of Ogden modelTab．4 The coeficients of Arruda-Boyce and Vander Waals models4．Stim less simulations4．1．Finite element modelingIn this paper，a Computer Aided Design(CAD)model of a rubber bushing was imported to the FEsoftware and then meshed to finite elements．Since agood model must be able to predict the rubber behav—ior in a large range of strains．the rubber part wasbuilt with a single continuum ，reduced-integrationand hybrid C3D8RH element．while the sleeve a—round the center hole of bush was built with S4 ele—merit．For the comparative analysis，seven kinds oflaws for isotropic rubber．1ike materials were employedrespectively．Due to the axisymmetrical structure ofrubber bushing and limited test equipments，the axialand radial stiffness were simulated．As shown in Fig.

3．the axial stiffness is along direction，and the ra—dial stiffness is along Y or direction.

Fig．3 CAD model of rubber bushingFig．4 shows the FE model of rubber bushingwith completely boundary conditions．To fuly assessthe models fitting．two lcad steps were set． In thefirst step，an imposed displacement was applied onthe outer surface of bushing．and simultaneously in-termediate shaft was fixed．This is a preloaded stepfor fixing the bushing and establishing nonlinear con‘tact steadily．In the second step，the lcads were ap·plied on the sleeve around the center hole，incremen—tally from 0．0 N to 1 000 N in direction and 0．0 Nt0 3 000 N in Y direction which enables the axial stiif-ness and the radial stiffness to be evaluated by FEsoftware respectively.

日 ，s∞0 们一日c_吕0ZO 5 O 5 O 5 O 3 ， 1 1 1 O ∞ 苣 ∞∞o 一日 g oZ 日 ∞0 ∽ _吕 0Z0 盯 ㈨S r 啪㈣N㈣2 O O LEI Gang，et al：A Study on Mechanical Behavior of Hyperelastic Materialfor Rubber Bushing 23Fig．4 FE model of rubber bushing4．2．Simulation resultsWith the laws for rubber．1ike material andboundary conditions having been established，the FEmodel may be evaluated．The simulations were car-ried out to compare with the two tests mentioned ear-lier。which are the radial stiffness test and the axialstiffness test．The compared results between the sim-ulated and experimental displacement—stress areshown in Fig．5 and 6，and a detailed test parame-ters list iS provided in Tab．5.

1 4001 2001 000800Z60040020001 400l 2001 000800Z60040020000 0 2 O．4 0．6 0 8 1．0 1．2 1 4 1．6 1．8 2．0inm 0 0．2 0．4 0．6 0．8 1．0 1．2 1．4 1．6 1．8 2．0m M Fig．5 The Load—displacement Curves at the axial load caseAs Tab．5 revealed，In general ，the FE modelusing the Van Der Waals constitutive equation fitsbeter．The good fitting in both radial stiffness andaxial stiffness was quantified as the relative error be.

tween simulation and experiment，and the erors are6．67％ and 3．45％ respectively． Good agreementbetween simulated and experimental data also appearsin the three．order Ogden and Yeoh data．As shown inTab．5．the range in relative error from 6％ to 8％ iSacceptable in engineering design or development.

However．an apparent deviation occurred in the Ar.

ruda．Boyce data． it shows that the Arruda—Boyceconstitutive model may be not good enough for therubber．1ike material at this case． From the abovementioned．Van der Waals iS the better constitutivemodel in terms of simulating the load—displacementcurves that can be used to evaluate the other staticstifness value which iS not obtained in experiment.

3 5003 0002 5002000Zl 500l 00050003 5003 0002 5002000Z1 5001 00050000 006 0 12 0．18 024 0．31 0．37 0．43 0．49 0．55m m 0 O．05 O10 015 O20 O．250．30 O350．400．45 O 50O．55m m Fig．6 The IJoad—displacement Curves at the radial load caseTab．5 Calculated stifness and simulation errors24 Hydromechatronics Engineering5．ConclusionsThe research results of the paper provide someinformation of the selection of constitutive models forthe rubber．1ike materia1．The results of fited Van derWaals model show a good agreement with materialtest results．In addition，by comparing the simulatedload—displacement with the experimental one， itshows that the Van der Waals model is likely a clea-rer superiority representation than other constitutivemodels.

Thus，the hyperealstic coefficients fiting proce-dure may be used in determining the constitutive pa.

rameters of rubber for suspension bushing．The entirework also offers a possible means to assess rubber—like materia】and select a suitable mode】for furtherstudy.

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[11]Yuan Qu，Linbo Zhang，ShenRong wu，et a1．Parame·ters Identification of Constitutive Models of RubberBushing[J]．SAE Technical Paper 2011—01—0795，2011．doi：10．4271／2011—01—0795.

1．重庆理工大学 a．汽车零部件制造及检测技术教育部重点实验室；b．机械工程学院，重庆 400054；2．西南交通大学 机械工程学院，成都 610031摘要：车辆悬架的橡胶衬套主要 由橡胶材料和被硫化在橡胶上的金属或塑料衬套组成，是一种影响车辆行驶平顺性、操纵稳定性以及NVH等性能的不可或缺的零件。研究了橡胶衬套特性，通过在ABAQUS和 HyperStudy中使用3种类型的材料试验数据，引入一种方法来对所选择的超弹性系数进行参数识别及拟合，建立橡胶衬套的高精度有限元模型，设置橡胶衬套的实际约束边界条件，通过仿真试验以得到相应径向与轴向刚度曲线。对仿真和真实试验结果进行对比，发现 Van der Waals模型具有比其他本构模型更优越的表现，针对提 出该某类衬套 ，该模型能更好地拟合其超弹性系数。

关键词：橡胶衬套；本构模型；材料试验；参数识别中图分类号：TH16 ‘